Linear Algebra

The study of linear algebra is fundamental for understanding and working effectively with numerical methods in MATLAB®. This branch of mathematics forms the basis of MATLAB®'s usage due to its powerful matrix and vector operations capabilities, enabling users to leverage the software's features fully. This chapter will delve into the essentials of linear algebra, including solving linear equations and manipulating matrices and vectors. We will explore the proper representation of linear equations and familiarize ourselves with the basic operations performed using matrix and vector notation. Additionally, we will review MATLAB® syntax for working with linear equations in matrix form. We will also address challenging and unsolvable problems and supplementary topics that enhance the practical application of algebraic concepts. In Sect. 3.1, a short introduction to the chapter is given. Section 3.2 describes the essential elements of matrices and vectors. Section 3.3 presents the implementation of different operations of matrices and vectors. Additionally, Sect. 3.4 reviews the most important features of systems of linear equations. Section 3.5 gives an overview of the determinant of a square matrix, and in Sect. 3.6, ill-conditioned problems are addressed. In Sects. 3.7 and 3.8, the implementation of Gauss elimination and Gauss‒Jordan elimination are reviewed, respectively. Section 3.9 LU decomposition is applied, and finally, in Sect. 3.10, the eigenvalues of matrices are discussed.